The workhorse of coated-bubble acoustics has long been the Rayleigh-Plesset equation and its many modified descendants. These models, extended over the years by researchers including de Jong, Sarkar, Tsiglifis and Pelekasis, and Paul, incorporate shell elasticity, viscous damping, buckling, rupture, and even shell compressibility and anisotropy. They are computationally cheap and analytically tractable, which explains their enduring popularity. But they all share a crippling assumption: spherical symmetry. A Rayleigh-Plesset-type model can only predict how the bubble radius changes in time. It cannot resolve nonspherical deformations, interfacial instabilities, jet formation, or the complex fluid-structure interactions that arise when a bubble is driven hard by ultrasound or sits close to a biological boundary such as a cell membrane.

An alternative family of approaches uses boundary element or boundary integral formulations to capture nonspherical dynamics. These methods have successfully modeled bubble breakup, transient saturation phenomena, three-dimensional contrast agent deformation, and even collapse and jet formation near a membrane during sonoporation. Yet they carry their own restriction: they assume potential flow in the surrounding liquid, which means liquid viscosity is neglected entirely. That assumption can be reasonable in some inertial cavitation regimes, but viscous effects become increasingly important in many biomedical ultrasound applications. Viscous damping of the surrounding medium can substantially influence bubble oscillations, and when a bubble oscillates or collapses near a stationary wall, solving the viscous flow with a no-slip condition naturally incorporates velocity gradients and wall shear stresses that potential-flow methods simply cannot see.

Fully resolved computational fluid dynamics offers a promising escape from both sets of limitations, because it can directly solve the surrounding multiphase flow without imposing spherical symmetry or inviscid assumptions. Garousi and Calvisi built their framework on two well-established numerical pillars. The first is the lattice Boltzmann method, a mesoscopic technique that represents fluid as fictitious particles propagating along discrete lattice directions and colliding at lattice sites. The method has emerged as an attractive framework for multiphase and interfacial flows owing to its parallel efficiency and suitability for handling complex moving interfaces. The second is the immersed boundary method, which represents deformable interfaces with discrete Lagrangian markers on an Eulerian fluid grid, coupling the two through velocity interpolation and force spreading, without requiring body-fitted meshes or interface reconstruction procedures.

The coupling of these two methods is what makes the new framework distinctive. The authors employed a multicomponent multiphase lattice Boltzmann solver based on the Shan-Chen pseudopotential approach, in which interaction forces between fluid components drive phase separation, combined with a Carnahan-Starling equation of state for the liquid to achieve realistic density contrasts. The solver was formulated in axisymmetric cylindrical coordinates, with the D2Q9 lattice stencil providing a suitable compromise between computational efficiency, isotropy, and accurate recovery of the Navier-Stokes equations. Onto this multiphase flow solver, the immersed boundary representation of the microbubble’s viscoelastic shell was grafted, enabling fully resolved two-way coupling between the surrounding fluid domains and the encapsulating shell within an acoustically driven environment.

The shell mechanics are described with care. Viscous effects within the shell are incorporated through the Boussinesq-Scriven constitutive law, a classical continuum mechanics framework for Newtonian interfaces that introduces interfacial shear and dilatational viscosities, allowing the shell to resist tangential deformation and surface area variation in a physically meaningful manner. Shell elasticity is integrated using an exponential elasticity model proposed by Paul and colleagues, chosen for its ability to capture the experimentally observed strain-softening behavior of ultrasound contrast agent shells under finite-area deformation. Acoustic forcing is introduced through a time-dependent far-field pressure condition representative of diagnostic and therapeutic ultrasound regimes, and the gas inside the bubble is modeled as perfluorobutane, the agent used in commercial contrast agents such as Sonazoid.

Validation proceeded in stages. First, the framework was tested against the Young-Laplace law, which dictates that the pressure jump across a curved interface in static equilibrium is proportional to the surface tension divided by the bubble radius. The solver reproduced the required linear relationship between pressure jump and inverse equilibrium radius with high accuracy, and a grid sensitivity study showed that refining the lattice spacing changed the predicted pressure jump by less than one percent. Critically, the authors demonstrated that the solver can handle physically relevant liquid-to-gas density ratios: by tuning the reduced temperature of the system, density ratios ranging from about 71 to 1387 were achieved, covering the order-of-1000 contrasts typical of air bubbles in water. Accurate representation of interfacial tension and multiphase equilibrium forms the foundation for reliable simulation of acoustically driven bubble dynamics, so these results matter enormously.

Next, the framework was validated against solutions of a modified Rayleigh-Plesset equation for encapsulated microbubbles driven by ultrasound, with shell parameters matching those measured for lipid-coated Sonazoid microbubbles. The simulations revealed that numerical accuracy depends strongly on the treatment of the immersed boundary’s interpolation-spreading stencil and the advection scheme used to move the Lagrangian markers. The lowest-order stencil produced the largest deviation from the theoretical solution, particularly near oscillation peaks where bubble expansion was overpredicted, while the fourth-order stencil yielded the closest match throughout. Similarly, among the first-order Euler, second-order Adams-Bashforth, and fourth-order Runge-Kutta advection schemes tested, the Runge-Kutta method provided the most accurate prediction of the spherical microbubble dynamics. Importantly, the framework remained accurate even in nonlinear acoustic regimes, where the bubble response departed substantially from sinusoidal behavior, exhibiting asymmetric expansion and compression phases, waveform distortion, and higher-harmonic features.

The shell property studies delivered some of the most physically rich results. Increasing shell elasticity substantially modified the nonlinear oscillatory response: stiffer shells strengthened the elastic restoring stresses opposing surface area expansion, suppressing early radial growth, enhancing mechanical stability, and prolonging sustained oscillations, because a stiffer shell stores a larger portion of the acoustic energy elastically, reducing irreversible collapse tendencies. Due to nonlinear elastic energy storage and the strain-softening behavior of the elasticity model, larger shell elasticities eventually promoted larger subsequent expansions after the initial oscillation stage. Shell dilatational viscosity, by contrast, introduced substantial damping: as it increased, oscillations became progressively attenuated and the response smoother and less nonlinear, because viscous dissipation within the shell converts part of the acoustic energy into dissipative losses. These trends are consistent with prior work by Mobadersany and Sarkar and by Church, providing independent support for the physical behavior predicted by the new solver.

The crowning demonstration is a simulation of jet formation and nonspherical collapse of an ultrasound-driven microbubble near a membrane, a problem of central interest in sonoporation. The computed bubble shapes agreed closely with boundary integral results from Mobadersany and Sarkar, but the new framework went further, resolving the full viscous flow field during collapse. The simulations showed how the nearby no-slip membrane restricts liquid motion on one side while the free-slip boundary permits stronger tangential motion on the other, creating an asymmetric pressure gradient that accelerates liquid above the bubble downward into a narrow, high-speed jet directed toward the membrane. In the final stage of collapse, the liquid jet rapidly penetrates the collapsing bubble and impinges on the membrane, with the liquid velocity adjacent to the membrane reaching approximately fifty times its value at the start of insonation. This high-speed microjet is recognized as the principal mechanism responsible for membrane permeabilization during sonoporation. The framework is admittedly more computationally demanding than reduced-order models, with a representative simulation taking roughly three hours on a desktop processor versus seconds for the modified Rayleigh-Plesset equation, but it unlocks physics that reduced-order models cannot access, and parallelization, well suited to lattice Boltzmann frameworks, could substantially cut runtimes. The authors point toward future extensions including shape mode oscillations, fully bidirectional bubble-boundary interactions with deformable membranes, fully three-dimensional simulations, and coupling with models for heat transfer, mass transport, or biological tissue mechanics, establishing a versatile computational platform for ultrasound-mediated drug delivery, mechanotransduction, cavitation-based tissue ablation, and other biomedical ultrasound phenomena involving complex multiphase fluid-structure coupling.

Subject of Research: Physics

Article Title: A computational framework based on immersed boundary-lattice Boltzmann coupling for ultrasound-excited encapsulated microbubbles

Article References: Garousi, M., & Calvisi, M. L. (2026). A computational framework based on immersed boundary-lattice Boltzmann coupling for ultrasound-excited encapsulated microbubbles. Frontiers in Acoustics, 4, Article 1887893. https://doi.org/10.3389/facou.2026.1887893

Image Credits: AI Generated

DOI: 10.3389/facou.2026.1887893

Keywords: biomedical ultrasound, encapsulated microbubbles, fluid-structure interaction, immersed boundary-lattice Boltzmann method, multiphase flow, nonlinear bubble dynamics, sonoporation, viscoelastic shell, computational fluid dynamics, drug delivery, computational, framework